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There is a proposal to place a satellite in orbit around planet Mars.

The satellite is to have an orbital time T equal to the length of a day on Mars. It can be shown that

T^{2}=k R^{3}

where R is the orbital radius of the satellite and k is a constant.

Mars has a mass of $6.4 \times 10^{23} \mathrm{~kg}$ . Show that, for Mars, k is about $9 \times 10^{-13} \mathrm{~s}^{2} \mathrm{~m}^{-3}$ .

$\frac{mv^2}{r}=\frac{GMm}{r^2}$ , hence $v=\sqrt{\frac{GM}{R}}$ . Also $v=\frac{2\pi R}{T}$ .

OR

$m\omega^2r=\frac{GMm}{r^2}$ , hence $\omega^2=\frac{GM}{R^3}$ .

uses either of the above to get $T^2=\frac{4\pi^2}{GM}R^3$

OR uses $k=\frac{4\pi^2}{GM}$ , $k=9.2\times10^{-13}\ /\ 9.3\times10^{-13}$

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IB Physics HL D 1 2 Universal Gravitation Topic Practice

Question 1